Skip to main content
Log in

Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

A new method for computing the topological entropy of a piecewise monotonic transformation on the interval is presented. It uses a transition matrix associated with the transformation. For this matrix we give a spectral theorem. This can be used for an estimation of the accuracy of the algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

REFERENCES

  1. S. L. Baldwin and E. E. Slaminka, Calculating topological entropy, J. Statist. Phys. 85:1017–1033 (1997).

    Google Scholar 

  2. N. J. Balmforth, E. A. Spiegel, and C. Tresser, Topological entropy of one-dimensional maps: Approximations and bounds, Phys. Rev. Lett. 72:80–83 (1994).

    Google Scholar 

  3. L. Block and J. Keesling, Computing the topological entropy of maps of the interval with three monotone pieces, J. Statist. Phys. 66 (1992).

  4. L. Block, J. Keesling, S. Li, and K. Peterson, An improved algorithm for computing topological entropy, J. Statist. Phys. 55:929–939 (1989).

    Google Scholar 

  5. P. Collet, J. P. Crutchfield, and J. P. Eckmann, Computing the topological entropy of maps, Comm. Math. Phys. 88:257–262 (1983).

    Google Scholar 

  6. N. Dunford and J. T. Schwartz, Linear Operators Part I (Interscience Publishers, New York, 1958).

    Google Scholar 

  7. P. Gora and A. Boyarsky, Computing the topological entropy of general one-dimensional maps, Trans. Am. Math. Soc. 323:39–49 (1991).

    Google Scholar 

  8. F. Hofbauer, The box dimension of completely invariant subsets for expanding piecewise monotonic transformations, Mh. Math. 121:199–211 (1995).

    Google Scholar 

  9. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy i, Israel J. Math. 34:213–237 (1979).

    Google Scholar 

  10. F. Hofbauer, On intrinsic ergodicity of piecewise monotonic transformations with positive entropy ii, Israel J. Math. 38:107–115 (1981).

    Google Scholar 

  11. F. Hofbauer, Piecewise invertible dynamical systems, Probab. Theory Relat. Fields 72:359–386 (1986).

    Google Scholar 

  12. F. Hofbauer, The structure of piecewise monotonic transformations, Ergod. Th. Dynam. Sys. 1:159–178 (1981).

    Google Scholar 

  13. M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings, Sudia Math. 67:45–63 (1980).

    Google Scholar 

  14. S. Newhouse and T. Pignataro, On the estimation of topological entropy, J. Statist. Phys. 72:1331–1351 (1993).

    Google Scholar 

  15. P. Raith, Continuity of the Hausdorff dimension for piecewise monotonic maps, Israel J. Math. 80:97–133 (1992).

    Google Scholar 

  16. P. Walters, An Introduction to Ergodic Theory (Springer Verlag, Berlin/Heidelberg/New York, 1982).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Steinberger, T. Computing the Topological Entropy for Piecewise Monotonic Maps on the Interval. Journal of Statistical Physics 95, 287–303 (1999). https://doi.org/10.1023/A:1004585613252

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1004585613252

Navigation